What transformation can I use to enable a relatively unbiased meta-analysis of $\alpha$ reliability? I want to meta-analyse reliability estimates. For simplicity, assume the reliability estimate in question is Cronbach's $\alpha$ (aka Guttman's $\lambda$-3). Various packages exist to do meta-analyses in general (e.g. the R 'meta' package), but do not seem to have an option for reliability. However, reliability estimates will not be distributed normally because they are capped at a maximum of 1, so treating them as normally distributed will introduce a bias.
A solution for the analogous problem for correlations is to use Fisher's r to z transformation.
So, what transformation can be performed on $\alpha$ to enable meta-analysis?
Note that this question is in many ways equivalent to asking, 'What is an approximate sampling distribution from which reliability estimates are drawn?'
EDIT:
The question was based upon the false assumption that there were no R packages that enabled meta-analysis of $\alpha$. This is not true (see my answer and @Wolfgang's comment), but may still be useful for anyone trying to understand how the transformation is performed.
 A: EDIT:
In fact, there is an R package that will do an appropriate transformation: the 'metafor' package. Details can be found at wviechtb.github.io/metafor/reference/escalc.html. Search for "Cronbach's alpha" for the appropriate section. Credit to @Wolfgang for this information.
ORIGINAL ANSWER:
Rodriguez and Maeda (2006)[1] offer a solution. They note that the ratio $$(1-r_\alpha)/(1-\rho_\alpha)$$ is an F distribution with degrees of freedom = $(n-1)$ and $(J-1)$, where $r_\alpha$ is the sample $\alpha$, $\rho_\alpha$ is the population $\alpha$, and J is the number of items[2,3]. Thus, "a nonlinear monotonic normalizing transformation of the sample coefficient alpha [is]: $$T_i=(1-r_\alpha)^{1/3}$$[4]
They point out that this transformation is biased, but it is less biased than Fisher's r to z transformation (and is also less biased than ignoring this issue entirely). They further note that the bias in Fisher's transformation is usually ignored[1,4], so (my interpretation) it is probably ok to ignore here as well.
Therefore, using this transformed $\alpha$ coefficient, the weighted mean transformed alpha can be computed with the formula: $$\overline{T}=\Sigma w_i T_i/\Sigma w_i$$ where $w_i=1/v_i$ and $$v_i=\frac{18J_i(n_i-1)(1-r_{\alpha i})^{2/3}}{(J_i-1)(9n_i-11)^2}$$ (See Hakstian & Whalen, 1976[4] for a derivation of this formula).
The variance of $\overline{T}$ is $\overline{v}=1/\Sigma w_i$ with a standard error of $\sqrt{\overline{v}}$, enabling calculation of a confidence interval.
These values can be transformed back into $\alpha$ with the formula: $$\hat{\rho}_\alpha=|1-\overline{T}^3|$$
where $\hat{\rho}_\alpha$ is the estimate of $\rho_\alpha$.
Rodriguez and Maeda[1] provide SPSS syntax to calculate these values, which may be helpful in creating code in your statistics program of choice.
References
[1] Rodriguez, M. C., & Maeda, Y. (2006). Meta-analysis of coefficient alpha. Psychological Methods, 11(3), 306–322. https://doi.org/10.1037/1082-989X.11.3.306
[2] eldt, L. S. (1965). The approximate sampling distribution of Kuder-Richardson reliability coefficient twenty. Psychometrika, 30(3), 357–370. https://doi.org/10.1007/BF02289499
[3] Kristof, W. (1963). The statistical theory of stepped-up reliability coefficients when a test has been divided into several equivalent parts. Psychometrika, 28(3), 221–238. https://doi.org/10.1007/BF02289571
[4] Hakstian, A. R., & Whalen, T. E. (1976). A K-Sample Significance Test for Independent Alpha Coefficients. Psychometrika, 41(2), 219–231.
